A liquid rocket engine with regeneratively cooled chamber walls and nozzles would definitely be one among those components experiencing the steepest thermal gradients (Chamber wall inner - $3000 K$ and the outer wall - $800 K$, within few millimetres).

How are the engines designed to withstand such significant temperature gradient without failure due to differential expansion? Is the effect simply minuscule considering the size of the engine? Wouldn't the extreme temperature lead to a significant expansion of the inner walls compared to the outer ones?

$\begingroup$From wikipedia: "Regenerative cooling is seldom used in isolation; film cooling,[5] curtain cooling,[6] transpiration cooling, radiation cooling are frequently employed as well." Combination with other cooling methods limit the steepest thermal gradients. Using copper (less sensitive to cracks due to low hardness and high ductility) for the inner chamber wall.$\endgroup$
– UweNov 24 '18 at 0:33

$\begingroup$@Uwe I feel that despite the use of copper inner lining, the strength would be very low during engine start, in case of Cryo engines, given a prechill run of the regen lines will reduce the temperature! I am looking forward to a more rigorous answer. Still, thanks for the comment..$\endgroup$
– karthikeyanNov 24 '18 at 3:08

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$\begingroup$Copper melts at 1358 K and boils at 2868 K, so chamber inner wall temperature should be much lower than 3000 K anyway.$\endgroup$
– UweNov 24 '18 at 17:49

$\begingroup$@Uwe thanks for pointing out the fallacy in my line of thought!$\endgroup$
– karthikeyanNov 24 '18 at 18:12